Details on the baby's gender, name and weight have not yet been reported.
A rep for Miller has not officially commented on the birth report.
Miller's pregnancy was revealed back in January.
"[Friends] were not surprised by the news of the pregnancy. They're really good together," a source said at the time.
"They just spent the holidays in Paris together after they announced the news in London."
The news was later confirmed by Miller's stepmother, Kelly Hoppen, who also nixed rumours that the actress was set to tie the knot.
Hoppen told London's Evening Standard newspaper: "Sienna is definitely not engaged or getting married, and she is pregnant."
Back in March, Miller spoke of her excitement at being pregnant.
"I'm feeling fine. It's all progressing nicely and it's very exciting," she told Britain's Vogue magazine.
"I feel very grounded, more than ever - it's a wonderful feeling. But all of a sudden, I've started to get cravings. I usually don't have a sweet tooth but these tarts I could eat all day long!
"I'm in my second trimester, so I'm I-can-talk-about-it pregnant," said the notoriously-private actress, adding, "I'm under strict instructions not to say anything as it would completely defeat the point of everything I've tried to achieve in the last eight years."
Miller and 26-year-old actor Sturridge first stepped out as a couple in March 2011.
It was also claimed today that Lily Allen is pregnant.
The 27-year-old singer and her hubby Sam Cooper are "over the moon" to be expecting again, reports Britain's The Mirror.
"Lily has started telling her friends and everybody in her close circle is so excited for her," said a source.
"It was a surprise but Lily and Sam are thrilled. The pair of them have loved every minute of parenthood so far and wanted to expand their brood quickly. So it's great timing."
Allen and Cooper became parents to daughter Ethel Mary last November.
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